To find the function \( r(t) \) representing the amount of money Philip has left to save at \( t \) months, we can begin by stating that:
- Philip's goal is to save $3000.
- The function \( s(t) = 500 \times 1.4^t \) represents the amount he has saved after \( t \) months.
The amount Philip has left to save, \( r(t) \), can be represented as:
\[ r(t) = 3000 - s(t) \]
Substituting the function \( s(t) \) into this equation gives us:
\[ r(t) = 3000 - 500 \times 1.4^t \]
To find out how many months it will take for Philip to save at least $3000, we need to solve for \( t \) when \( s(t) \) is equal to or greater than $3000:
\[ 500 \times 1.4^t \geq 3000 \]
Dividing both sides by 500:
\[ 1.4^t \geq 6 \]
Now, we take the logarithm of both sides (using natural logarithm, but common logarithm would work just as well):
\[ \log(1.4^t) \geq \log(6) \]
Using the power rule of logarithms:
\[ t \cdot \log(1.4) \geq \log(6) \]
Now solve for \( t \):
\[ t \geq \frac{\log(6)}{\log(1.4)} \]
Calculating the values:
\[ \log(6) \approx 0.7781 \] \[ \log(1.4) \approx 0.1461 \]
Now, divide these values:
\[ t \geq \frac{0.7781}{0.1461} \approx 5.33 \]
Since \( t \) must be a whole number, we take the next whole number:
\[ t \geq 6 \]
Thus, after 6 months, Philip will have saved at least $3000.